Seminars and Colloquia by Series

A new type of exceptional Laguerre polynomials

Series
Analysis Seminar
Time
Monday, March 10, 2014 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 171
Speaker
Conni Liaw – Baylor University
The Bochner Classification Theorem (1929) characterizes the polynomial sequences $\{p_n\}_{n=0}^\infty$, with $\deg p_n=n$ that simultaneously form a complete set of eigenstates for a second order differential operator and are orthogonal with respect to a positive Borel measure having finite moments of all orders: Hermite, Laguerre, Jacobi and Bessel polynomials. In 2009, G\'{o}mez-Ullate, Kamran, and Milson found that for sequences $\{p_n\}_{n=1}^\infty$, with $\deg p_n=n$ (i.e.~without the constant polynomial) the only such sequences are the \emph{exceptional} Laguerre and Jacobi polynomials. They also studied two Types of Laguerre polynomial sequences which omit $m$ polynomials. We show the existence of a new "Type III" family of Laguerre polynomials and focus on its properties.

Graphs with the maximum number of proper q-colorings

Series
Combinatorics Seminar
Time
Friday, March 7, 2014 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Jie Ma – Carnegie Mellon
We study an old problem of Linial and Wilf to find the graphs with n vertices and m edges which maximize the number of proper q-colorings of their vertices. In a breakthrough paper, Loh, Pikhurko and Sudakov asymptotically reduced the problem to an optimization problem. We prove the following structural result which tells us how the optimal solutionlooks like: for any instance, each solution of the optimization problem corresponds to either a complete multipartite graph or a graph obtained from a complete multipartite graph by removing certain edges. We then apply this result on optimal graphs to general instances, including a conjecture of Lazebnik from 1989 which asserts that for any q>=s>= 2, the Turan graph T_s(n) has the maximum number of q-colorings among all graphs with the same number of vertices and edges. We disprove this conjecture by providing infinity many counterexamples in the interval s+7 <= q <= O(s^{3/2}). On the positive side, we show that when q= \Omega(s^2) the Turan graph indeed achieves the maximum number of q-colorings. Joint work with Humberto Naves.

The subadditive ergodic theorem

Series
Dynamical Systems Working Seminar
Time
Friday, March 7, 2014 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 05
Speaker
Mikel J. de Viana – Georgia Tech
We will present a proof of the subadditive ergodic theorem. This is part of a reading seminar geared towards understanding of Smooth Ergodic Theory. (The study of dynamical systems using at the same time tools from measure theory and from differential geometry)It should be accesible to graduate students and the presentation is informal. The first goal will be a proof of the Oseledets multiplicative ergodic theorem for random matrices. Then, we will try to cover the Pesin entropy formula, invariant manifolds, etc.

Cancelled

Series
Stochastics Seminar
Time
Thursday, March 6, 2014 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Ioana Dumirtiu – Univ. of Washington

Mark Kac's Master Equation and Propagation of Chaos

Series
SIAM Student Seminar
Time
Thursday, March 6, 2014 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Hagop Tossounian – School of Mathematics, Georgia Tech
In 1956 Mark Kac introduced an equation governing the evolution of the velocity distribution of n particles. In his derivation, he assumed a stochastic model based on binary collisions which preserves energy but not momentum. In this talk I will describe Kac's model and the main theorem of Kac's paper : that solutions with chaotic initial data can be related to the solutions Boltzmann type equation.

Riemann-Roch theory via partial graph orientations

Series
Graph Theory Seminar
Time
Thursday, March 6, 2014 - 12:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Spencer Backman – Math, GT
This talk is a sequel to the speaker's previous lecture given in the January 31st Combinatorics Seminar, but attendance at the first talk is not assumed. We begin by carefully reviewing our generalized cycle-cocyle reversal system for partial graph orientations. A self contained description of Baker and Norin's Riemann-Roch formula for graphs is given using their original chip-firing language. We then explain how to reinterpret and reprove this theorem using partial graph orientations. In passing, the Baker-Norin rank of a partial orientation is shown to be one less than the minimum number of directed paths which need to be reversed in the generalized cycle-cocycle reversal system to produce an acyclic partial orientation. We conclude with an overview of how these results extend to the continuous setting of metric graphs (abstract tropical curves).

Automorphisms of Drinfeld's half-spaces over a finite field

Series
School of Mathematics Colloquium
Time
Thursday, March 6, 2014 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Annette Werner – Johann Wolfgang Goethe-Universität (Frankfurt)
Drinfeld's upper half-spaces over non-archimedean local fields are the founding examples of the theory of period domains. In this talk we consider analogs of Drinfeld's upper half-spaces over finite fields. They are open subvarieties of a projective space. We show that their automorphism group is the group of automorphisms of the ambient projective space. This is a problem in birational geometry, which we solve using tools in non-archimedean analytic geometry.

Buildings and Berkovich spaces

Series
Algebra Seminar
Time
Wednesday, March 5, 2014 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 254
Speaker
Annette Werner – Johann Wolfgang Goethe-Universität (Frankfurt)
The goal of this talk is to show that Bruhat-Tits buildings can be investigated with analytic geometry. After introducing the theory of Bruhat-Tits buildings we show that they can be embedded in a natural way into Berkovich analytic flag varieties. The image of the building is contained in an open subset which in the case of projective space is Drinfeld's well-known p-adic upper half plane. In this way we can compactify buildings in a natural way.

The Pick Problem and Related Function Spaces

Series
Research Horizons Seminar
Time
Wednesday, March 5, 2014 - 12:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Dr. Bickel – School of Math
The classic Pick Interpolation Problem asks: Given points z_1, z_n and w_1, w_n in the unit disk, is there a function f(z) that (1) is holomorphic on the unit disk, (2) satisfies f(z_i)=w_i, and (3) satisfies |f(z)|=1 In 1917, Pick showed that such a function f(z) exists precisely when an associated matrix is positive semidefinite. In this talk, I will translate the Pick problem to the language of Hilbert function spaces and present a more modern proof of the Pick problem. The benefit of this approach is that, as shown by J. Agler in 1989, it generalizes easily to the two-variable setting. At the heart of the proof is a method of representing bounded analytic one and two-variable functions using Hilbert space operators. Time-permitting, I will discuss recent results concerning the structure of such representations for bounded two-variable analytic functions, which is joint work with G. Knese.

Systems Biology of Epidemiology: From Genes to Environment

Series
Mathematical Biology Seminar
Time
Wednesday, March 5, 2014 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Professor Juan Gutierrez – UGA –
The traditional epidemiological approach to characterize transmission of infectious disease consists of compartmentalizing hosts into susceptible, exposed, infected, recovered (SEIR), and vectors into susceptible, exposed and infected (SEI), and variations of this paradigm (e.g. SIR, SIR/SI, etc.). Compartmentalized models are based on a series of simplifying assumptions and have been successfully used to study a broad range of disease transmission dynamics. These paradigm is challenged when the within-host dynamics of disease is taken into account with aspects such as: (i) Simultaneous Infection: An infection can include the simultaneous presence of several distinct pathogen genomes, from the same or multiple species, thus an individual might belong to multiple compartments simultaneously. This precludes the traditional calculation of the basic reproductive number. (ii) Antigenic diversity and variation: Antigenic diversity, defined as antigenic differences between pathogens in a population, and antigenic variation, defined as the ability of a pathogen to change antigens presented to the immune system during an infection, are central to the pathogen's ability to 1) infect previously exposed hosts, and 2) maintain a long-term infection in the face of the host immune response. Immune evasion facilitated by this variability is a critical factor in the dynamics of pathogen growth, and therefore, transmission.This talk explores an alternate mechanistic formulation of epidemiological dynamics based upon studying the influence of within-host dynamics in environmental transmission. A basic propagation number is calculated that could guide public health policy.

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